Identify Valid Formulas for Calculating the Area of a Circle

Question

Which formulas can be used to find the area of a circle? Choose ALL that apply: $A = \pi(2r)^2$, $A = 2\pi r$, $A = \pi\left(\frac{1}{2}d\right)^2$, $A = 2\pi d$, $A = \pi r^2$, $A = \pi d$

Accepted Answer

$A = \pi\left(\frac{1}{2}d\right)^2$, $A = \pi r^2$

Answer

Step 1: Recall the standard formula for the area of a circle: $A = \pi r^2$, where $r$ is the radius of the circle. This confirms $A = \pi r^2$ is a valid formula. Step 2: Relate radius $r$ to diameter $d$. Since the diameter is twice the radius, $r = \frac{1}{2}d$. Step 3: Substitute $r = \frac{1}{2}d$ into the standard area formula: $A = \pi\left(\frac{1}{2}d\right)^2$. This shows $A = \pi\left(\frac{1}{2}d\right)^2$ is equivalent to the standard formula and also valid. Step 4: Eliminate incorrect options: $A = \pi(2r)^2$ incorrectly uses $2r$ (diameter) as the radius in the area formula; $A = 2\pi r$ and $A = 2\pi d$ are circumference formulas; $A = \pi d$ is neither a valid area nor circumference formula.